An infinite number of charges, each of charge 1 μ μ C, are placed on the x-axis with co-ordinates x = 1, 2, 4, 8, .... ∞ ∞ . If a charge of 1 C is kept at the origin, then what is the net force acting on 1 C charge
Text Solution
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The schematic diagram of distribution of charges on x-axis is shown in figure below:
Total force acting on 1 C charge is given by
$F = \frac{1}{4 \pi \varepsilon_0} \left[ \frac{1 \times 1 \times 10^{-6}}{(1)^2} + \frac{1 \times 1 \times 10^{-6}}{(2)^2} \right]$
$+ \frac{1 \times 1 \times 10^{-6}}{(4)^2} + \frac{1 \times 1 \times 10^{-6}}{(8)^2} + \ldots \infty ]$
$= \frac{10^{-6}}{4 \pi \varepsilon_0} \left( \frac{1}{1} + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots^{\infty} \right) = 9 \times 10^{9} \times 10^{-6} \left( \frac{1}{1 - \frac{1}{4}} \right)$
$= 9 \times 10^{9} \times 10^{-6} \times \frac{4}{3} = 9 \times 10^{3} \times \frac{4}{3}^{4}$ = 12000 N
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